Parametric Method To Design Functional Observer For Linear Time-varying Systems

Due to the difficulty of direct measurement,the economy and operability of equipment,it is not practical to obtain all state variables in many practical systems.Luenberger put forward the problem of observer design based on this situation,which has been widely concerned by experts and scholars.So far,most studies have focused on linear time-invariant systems.When the parameters of the system change rapidly with time,such as the spacecraft performing attitude maneuver,orbital rendezvous and earth observation tasks,the changes of the surrounding environment of the system will cause its operating conditions to fluctuate within a large range,so the requirements for performance indicators will be very high.At this point,a time-varying model can achieve the control goal more accurately.Therefore,it is very necessary to study the linear time-varying system.In this paper,the linear time-varying system is taken as the research object,through the analysis of its related stability,aiming at the problem that the state variables are not easy to obtain,the dynamic equations of the reduced-order functional observer and the functional interval observer of the first-order system,the functional observer of the second-order system are established respectively,and the stability of the observation error system is used to obtain conditions for the existence of the observer.Further,the observer design problem is transformed into solving a class of time-varying Sylvester matrix equations,and a feasible design scheme is presented based on parameterization method.It mainly includes the following aspects:(1)For the first-order linear time-varying system,the construction principle of the reduced-order functional observer is given by non-singular transformation based on the partial state information of the system output,and then the solution method of the observer design problem is given based on the parameterization method.By analyzing the stability of error dynamic system,sufficient criteria for the estimation performance of observer system are given.Further,the above criterion is translated into the problem of solving the time-varying Sylvester matrix equation,and a parametric method for the design of the reduced-order functional observer is proposed,and a detailed design flow is given.Compared with the existing design methods,the parametric design is less conservative and simple to implement,making it easy to apply in engineering.(2)For a class of first-order linear time-varying systems with state and output perturbations,a novel functional interval observer and its design parameterization method are proposed.First,the sufficient conditions for the existence of the functional interval observer are proposed by using the positive system theory to ensure the non-negativity of the error system.Further,combining the above sufficient conditions with the solution of the time-varying Sylvester matrix equation,the parametric expressions for the gain matrices of the observer are proposed,and a simple and effective scheme for solving the interval estimation problem of a first-order linear time-varying system is also given.Finally,the effectiveness of the proposed method is verified by a bank-to-turn(BTT)missile system.(3)For the second-order linear time-varying system,the corresponding functional signal observer is designed using a parametric approach,and the state vector of the second-order system is reconstructed.First,the functional observer is built directly in the framework of the second-order time-varying system by drawing on the theory of the observer in the second-order constant case,and the conditions for the existence of this observer are discussed based on the stability analysis of the error system.Then the parametric design method of the observer is given by solving a set of time-varying Sylvester matrix equations with constraints,and the free parameters present in the method provide degrees of freedom for the observer design,which can be utilized to meet other performance requirements of the system.In addition,for second-order time-varying systems,this parametric design method eliminates the limitations of the conventional method(transformed into a first-order incremental system to handle it),and the designed observer can perform state estimation at the desired convergence rate,which greatly improves the estimation performance.Finally,the simulation results demonstrate the simplicity and effectiveness of the proposed method.